need to solve limits question

Question

need to solve this using laws from limits unit in calculus 12 

---

---

---

Answer 1

(a) Factorise: (2x+5)(x+1)/((x+3)(x+4). No common factors between numerator and denominator.

As x→-3, denominator→0, but numerator evaluates to 2, so limit is undefined (infinity).

(b) Factorise: (x-2)(x+2)/(x(x-2)). When x=-2, numerator=0, denominator≠0. No anomalies, so limit is zero.

(c) Factorise: (x-3)(2x+1)/((x-3)(x+3)). Common factor x-3 shared between numerator and denominator. The expression becomes (2x+1)/(x+3) except for x=3. But as x→3, the expression→7/6. Therefore the limit is 7/6.

(d) The denominator can be written (125-x³)/(125x³) making the whole expression, after factoring:

125x³(x-5)/((5-x)(25+5x+x²). x-5 is a common factor reducing the expression when x≠5:

-125x³/(25+5x+x²). When x→5, limit→-625/3.